Immersions in the metastable dimension range via the normal
bordism approach
C. Biasi
a,∗, D.L. Gonçalves
b,1, A.K.M. Libardi
caDepartamento de Matemática, ICMC-USP, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil bIME-USP, Caixa Postal 66281, Agência Cidade de S. Paulo, 05315-970, São Paulo, SP, Brazil
cDepartamento de Matemática, IGCE-UNESP, Bela Vista, 13506-700, Rio Claro, SP, Brazil
Received 19 October 1999; received in revised form 10 April 2000
Abstract
Let f : M→ N be a continuous map between two closed n-manifolds such that f∗: H∗(M,Z2)→ H∗(N ,Z2) is an isomorphism. Suppose that M immerses inRn+k for 5 n < 2k. Then N also
immerses inRn+k. We use techniques of normal bordism theory to prove this result and we show that for a large family of spaces we can replace the homology condition by the corresponding one in homotopy.2001 Elsevier Science B.V. All rights reserved.
AMS classification: Primary 57R42, Secondary 55Q10; 55P60
Keywords: Normal bordism; Immersion of manifold; Localization; Group action; Nilpotent space
1. Introduction
In this paper we are concerned with the following immersion problem:
Let M and N be closed smooth connected n-dimensional manifolds and let f : M→ N be a continuous map. Suppose that M immerses inRn+k, for some k, with 5 n < 2k. Under which conditions on f does N immerse inRn+k?
For the special case where f is a homotopy equivalence between M and N , if M immerses inRn+k for some k [n/2] + 2, Glover and Mislin in [5] proved that N also immerses in Rn+k. They also proved that in the case where M and N are connected simple2orientable and closed smooth manifolds and the 2-localizations M2 and N2 are
*Corresponding author.
E-mail addresses: biasi@icmc.sc.usp.br (C. Biasi), dlgoncal@ime.usp.br (D.L. Gonçalves),
alicekml@ms.rc.unesp.br (A.K.M. Libardi).
1The second author has been partially supported by CNPq. 2A manifold X is simple if π
1(X) acts trivially on π∗(X).
0166-8641/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 6 - 8 6 4 1 ( 0 0 ) 0 0 0 9 3 - 6
homotopy equivalent, if M immerses inRn+k for some k [n/2] + 1, then N immerses inRn+2[k/2]+1.
Later, Glover and Homer in [3,4] proved the same result under weaker hypotheses on M and N , for example, nilpotent manifolds. In fact, in [3] they considered the more general problem where M immerses in another manifold.
In our work, the manifolds M and N are not required to be nilpotent. Concerning the dimension of the space that M and N immerse into, our results are improvement of the results in [5] by 1, and are the same as the ones from [4]. Nevertheless we do not require
Hk(M,Q) = 0, as well k be an odd number, as it is required in [4]. We use a normal
bordism approach to investigate this problem. We prove the following main results:
Theorem A. Let M and N be closed smooth connected n-manifolds and let f : M→ N
be a continuous map such that f∗: Hi(M,Z2)→ Hi(N,Z2)
is an isomorphism for i 0.
Then if M immerses inRn+kfor 5 n < 2k, so does N.
Theorem B. Let f : M→ M be a finite regular covering of a closed connected smooth n-manifold M.
Suppose π = π1(M)/f#π1( M) is of odd order and operates trivially on H∗( M,Z2).
Then if M immerses inRn+kfor 5 n < 2k, so does M.
In the next theorem, for f : M→ N, πi(N, M) means πi(Zf, M), where Zf is the
mapping cylinder of f .
Theorem C. Let M and N be closed smooth connected n-manifolds, f : M → N
a continuous map and H = f#π1(N ). Suppose the inclusion i : H → π1(N ) induces
isomorphisms in homology with Z2 coefficients, the index[π1(N ), H] is finite and odd,
and πi(N, M) is an odd torsion group for 1 < i n. Then if M immerses in Rn+k for
5 n < 2k, so does N.
The work is divided in four sections. In Section 2 we review the normal bordism theory and in Section 3 we prove Theorems A and B. In Section 4 we show that, for a family of spaces, larger than the family of nilpotent spaces, we can replace the homology hypotheses given in Theorem A by the corresponding one in homotopy. Finally, in Section 5 we analyze algebraic conditions in terms of the fundamental group of the spaces, in order to have an isomorphism of homology with Z2 coefficients and then we prove Theorem C. We finish the section by giving an example where we have an isomorphism in homotopy modulo finite odd groups, but not an isomorphism of homology with Z2 coefficients.
2. Normal bordism
Given a topological space X and a virtual bundle φ, Ωi(X, φ) denotes the ith normal
bordism group of X with coefficient φ. We adopt the Salomonsen convention.
Let M be a closed smooth n-manifold and X a connected smooth (n+ k)-manifold. Let us consider h : M→ X a map with dim ν = k + , for large. Here ν denotes the stable normal bundle of h.
Then, by Hirsch [9], h is homotopic to an immersion if and only if there is a monomorphism from M× Rto ν or equivalently geom dim(ν) k.
In [13,14], Koschorke defines an invariant ωk(ν)∈ Ωn−k−1(M× P∞, φ) which is an
obstruction to the existence of a monomorphism from M× Rinto ν.
We recall that φ= λ ⊗ (ν − ε)− T M is a virtual vector bundle over M × P∞, where λ denotes the canonical line bundle over the real projective space P∞. For the definition and more details about normal bordism see [12] or [17].
So geom dim(ν) k if and only if ωk(ν)= 0, provided n < 2k. See [13,14].
Salomonsen defines a fibre bundle πM: Vk(Ψ )→ M such that the existence of a cross
section s : M→ Vk(Ψ ) implies that geom dim(Ψ ) k. Here Ψ is a virtual bundle over M.
In order to study whether cross sections exist we consider (πM)∗: Ωn(Vk(Ψ ), T M0)→ Ωn(M, T M0), where T M0 is the virtual vector bundle T M − εn. We note that if geom dim(Ψ ) k then (πM)∗is onto. We can also consider the following exact sequence,
for 5 n < 2k [17]. · · · → ΩnVk(Ψ ), T M0 (πM)∗ −→ Ωn M, T M0−→ ΩγM n−k−1 M× P∞, φ→ · · · . (I)
The mapping γM is defined by the construction of the sequence and
φ= −(n − k − 1)λ − λ ⊗ Ψ + T M0.
Let[M] = [M, 1M, tM] ∈ Ωn(M, T M0) be the fundamental class of M where tM: T M⊕ εn∼= εn⊕ T M
is the isomorphism which interchange factors. If we consider Ψ = h T X− εk⊕ T M, then
we have that γM[M] = ωk(ν).
For the next lemma we recall that F (q) is the monoid of degree 1 or −1 pointed maps of Sq and F =F (q). For a connected finite CW complex X, let α∈ [X, BF]
be a stable bundle over X and we define αp as the composition of α and the canonical
map BF→ (BF)p, where (BF)p is the p-localization of BF (p prime or 0). The map α
represents a fibre bundle π : E→ M with fibre Sr, r large, and αp represents the fibre bundle (π )p: (E)p→ (M)pwith fibre (Sr)p. The Thom complexes T (α) and T (αp) are
respectively the mapping cones of π and πp.
LetC denote the class of all torsion groups where the torsion is odd.
Lemma 2.1. Let f : M→ N be a mapping between closed connected smooth n-manifolds
such that
is an isomorphism for i 0. Then we have:
(1) f∗: Ωn(M, f∗T N0)→ Ωn(N, T N0) is aC-isomorphism.
(2) f∗(β2)= α2, where α= νM and β= νNare the stable normal bundles over M and N respectively.
Proof. The first part of this lemma is a special case of Lemma 2 in [17].
For the second part, let us define θ∈ [M, BF] by θp= αp, p= 2, and θ2= f∗(β2). We observe that for p= 2, T (θp) is S-reducible, because αp= (νM)p.
We now use the same techniques of [5] in the proof of their Proposition 4.1, but applied to the Thom complexes. We point out that since f∗ is an isomorphism for i 0, we have homotopy equivalences between the Thom complexes (T (f∗β))2and (T (β))2. But (T (f∗β))2= (T (f∗β2))2= (T (θ2))2= (T (θ))2and since T (β) is S- reducible, it follows that T (θ ) is S-reducible at 2. Therefore T (θ ) is S-reducible and by Proposition 5.6 [18], we have that θ= α. ✷
3. Proofs of Theorems A and B
Proof of Theorem A. Let us consider the following commutative diagram
Ωn(Vk(ΨM), f∗T N0) G∗ Ωn(Vk(ΨN), T N0) (πM )∗ (πN)∗ Ωn(M, f∗T N0) f∗ Ωn(N, T N0) γM γN Ωn−k−1(M× P∞, φM) F∗ Ωn(N× P∞, φN)
where the vertical exact sequences are obtained from (I) and G∗and F∗are given in [17],
ΨM= εn+k− T M ⊕ εkand ΨN= εn+k− T N ⊕ εk. We observe that (πM )∗is the induced
map of πM in normal bordism groups with virtual bundle f∗T N0.
Since f∗is aC-isomorphism by Lemma 2.1, there exists an odd number 1such that 1· [N] = f∗(x), for some x∈ Ωn
M, f∗T N0.
If we prove that (πM )∗is aC-epimorphism, then there exists an odd number 2such that 2· x = (πM )∗(y) where y∈ Ωn(Vk(ΨM), T M0).
Using the commutative diagram we have that (πN)∗G∗(y)= f∗(πM )∗(y)= f∗(2·x) = 2· 1[N] and so 2· 1· γN[N] = 0.
Since the elements of the image of γNhave order a power of 2, by Theorem 10.2 in [17], and 2· 1is an odd number, γN[N] = 0 and N immerses in R
n+k.
Thus, all we need to prove is that if M immerses in Rn+k, then (πM )∗ is a C-epimorphism and this is what we will do.
We recall that α= νM and β= νN, as in the Lemma 2.1.
We use the following commutative diagram, equivalent to the top commutative diagram above, ΠnS+p(T (f∗β)) G∗ ΠnS+p(T (β)) (πM )∗ (πN)∗ ΠnS+p(T (f∗β)) f∗ ΠnS+p(T (β))
where β denotes the pull back of β by πN. We observe that T (f∗β)2=T (f∗β2) 2= T (α2) 2= T (α)2,
where the second equality is given by Lemma 2.1. We also have that (T (f∗β))2 = (T (α))2, whereα denotes the pullback of α by πM.
Now, if M immerses inRn+kthen
(πM)∗: ΩnVk(ΨM), T M0
→ Ωn
M, T M0
is an epimorphism and then
ΠnS+p(T (α))2 → ΠS n+p (T (α))2
is aC-epimorphism or equivalently ΠnS+p((Tf∗β)2)→ ΠnS+p((Tf∗β)2) is a C-epimor-phism and the result follows. ✷
Proof of Theorem B. If π has odd order and since the coefficient group of the homology
group isZ2, the divisibility condition of Theorem 10.8.8 in [11] is satisfied. Therefore H∗( M,Z2)≈ H∗(M,Z2) and the result follows by Theorem A. ✷
4. Applications
Throughout this paragraph we will consider M, N connected closed and smooth n-manifolds and f : M→ N a continuous map.
We consider the following cases: (1) f is a homotopy equivalence.
(2) M and N are nilpotent orientable and f is such that the induced f2: M2→ N2, from the 2-localizations M2of M into N2of N , is a homotopy equivalence.
(3) M and N areC-nilpotent manifolds and f is such that
f#1: π1(M)→ π1(N ) is aC-epimorphism and
f#i: πi(M)→ πi(N ) is aC-isomorphism
for i < n and aC-epimorphism for i = n. For details about nilpotent spaces see [10] and forC-nilpotent spaces see [7].
(4) f : M→ N is such that
f#1: π1(M)→ π1(N ) is an epimorphism and
f#i: πi(M)→ πi(N ) is aC-isomorphism for i < n
and aC-epimorphism for i = n.
(5) f is an orientation true map (i.e., the loops α and f#1(α) have the same sign for all α∈ π1(M)) such that
f∗: Hi(M, Z)→ Hi(N, Z)
is an isomorphism for i 0, where Z denotes the twisted integer coefficients over
M respectively N, associated to w1(M) respectively w1(N ).
In all the cases above we have that if M immerses inRn+kfor 5 n < 2k, so does N. Under the hypotheses of cases (1) or (2), the result follows immediately from Theorem A. For the third and the fourth cases, we use the Proposition 3.5 in [7] and Theorem 1.1 in [2] respectively in order to show that f∗: Hi(M,Z2)→ Hi(N,Z2) is an isomorphism for i 0. So, by Theorem A, the result follows.
The last case follows from Theorem 3.5 in [1]. Theorem B has an interesting consequence:
Corollary 4.1. If Hi(M,Z2)= 0 or Z2, for all i, and if G is an odd order group which
acts freely on M, then if M immerses inRn+k for 5 n < 2k, so does M/G.
5. The mod 2 homology isomorphism condition
In this section we will assume that fi#·πi(M)→ πi(N ) satisfies the hypotheses: fi#is a C-isomorphism for 1 < i < n, C-epimorphism for i = n, C-injective for i = 1 and the index
[π1(N ), f1#(π1(M))] is finite and odd. We will study the question of how to decide when a map f : M→ N induces an isomorphism fi∗: Hi(M,Z2)→ Hi(N,Z2) for 0 i n.
We will also provide an example where this does not happen. As before,C stands for the class of all torsion groups where the torsion is odd.
Proposition 5.1. Given f : M → N as above, there is a finite cover p : N → N and a
lifting
N
M f
f N
such that f∗i: Hi(M,Z2)→ Hi(N ,Z2) is an isomorphism for i 0.
Proof. Let N be the cover which corresponds to the subgroup f1#(π1(M)) and f a lift of f . By Theorem 1.1 of [2] the result follows. ✷
Corollary 5.2. Suppose that f#(π1(M)) is a normal subgroup of π1(N ). If Hi(M,Z2)= 0
orZ2, for all i, then if M immerses inRn+k for 5 n < 2k, so does N.
Proof. The normal subgroup condition implies that the covering is regular [11]. The result
follows now from Proposition 5.1 and Theorem B. ✷
For the next proposition let us consider the commutative diagram
N p N N p N K(π1(N ), 1) p K(π1(N ), 1)
where N , N are the universal covers of N , N respectively, and K(π, 1) are Eilenberg–
MacLane spaces.
Proposition 5.3. The induced homomorphisms pi∗: Hi(N ,Z2)→ Hi(N,Z2) are
isomor-phisms for all i 0 if pi∗: Hi(π1(N ),Z2)→ Hi(π1(N ),Z2) is an isomorphism for all i 0, where H∗means group homology.
Proof. Since N and N are simply connected andp#: πi(N )→ πi( N ) is aC-isomorphism
for i 0, we have that pi∗: Hi(N ,Z2)→ Hi( N ,Z2) is an isomorphism. Now we consider the Serre spectral sequence of the fibrations and the induced map. The result follows from the spectral mapping theorem. ✷
From now on let us denote H = π1(N ), G= π1(N ) and j : H→ G the inclusion. We have that the index[G, H ] is odd.
Proposition 5.4. The homomorphism ji∗: Hi(H,Z2)→ Hi(G,Z2) is an epimorphism for i 0.
Proof. It is equivalent to show that the induced map in cohomology is injective. For this we
consider the composite of the corestriction with the restriction (see Chapter II, Sections 2.3 and 2.4 in [19]) τ : Hi(G,Z2)→ Hi(H,Z2)→ Hi(G,Z2), which is multiplication by = [G, H ]. Since is odd this is an isomorphism. So it follows that the first map is
injective and hence the result. ✷
Corollary 5.5. If G is finite of odd order, then Hi(G,Z2)= 0, i > 0.
Proof. We take H the trivial group. By Proposition 5.3 Hi(G,Z2)= 0, i > 0 . ✷
Proof of Theorem C. The result follows directly from Propositions 5.1, 5.2 and
Theorem A. ✷
In general one can not expect to have ji∗: Hi(H,Z2)→ Hi(G,Z2) an isomorphism for all i > 0.
Example 5.6 (Algebraic). Let us consider the following extension
0→ Z ⊕ Z ⊕ Z → G → Z3→ 0
where the action ω :Z3→ Aut(Z ⊕ Z ⊕ Z) is given by
ω(1)= −1 −1 0 1 0 0 0 0 1
and the extension corresponds to a nontrivial element of H2(Z3,Z ⊕ Z ⊕ Z) ≈ Z3. By using Chapter IV, Section 7 [15], it is a standard calculation that H2(Z3,Z ⊕ Z ⊕ Z) Z3. So let H= Z ⊕ Z ⊕ Z.
We will use the Lyndon–Hochschild–Serre spectral sequence, in order to compute
H∗(G,Z2). The E2term is Ep,q2 = Hp(Z3, Hq(Z ⊕ Z ⊕ Z; Z2)). So Ep,q2 = 0 q 4; Hp(Z3,Z2) q= 0, 3; Hp(Z3,Z2⊕ Z2⊕ Z2) q= 1; Hp(Z3,Z2⊕ Z2⊕ Z2) q= 2.
Where the local coefficient is given by ω1(1)= 1 1 0 1 0 0 0 0 1 and ω2(1)= 1 0 0 0 1 1 0 1 0 ,
for the cases q= 1, q = 2, respectively.
Again, using [15] Chapter IV, Section 7, we get
Hp(Z3,Z2⊕ Z2⊕ Z2)= Z2 p= 0, 0 p= 0,
where the action ω is either ω1or ω2. So we get
H∗(G,Z2)= Z2 ∗ = 0, 3; H1(H ,Z2,⊕Z2⊕ Z2)≈ Z2; H2(H ,Z2,⊕Z2⊕ Z2)≈ Z2. Remarks.
• It is not difficult to see that the group G is not nilpotent but it is certainly infra-abelian. • Let ω : Q → SL(n, Z) be an action. It is natural to look for weaker hypothesis than the
one that ω is trivial which still implies that Hi(Z ⊕ · · · ⊕ Z n
,Z2)→ Hi(G,Z2) is an isomorphism for all i, where G is any extension ofZ ⊕ · · · ⊕ Z
n
by Q∈ C with action
ω. The condition that the reduced action ω2: Q→ SL(n, Z2) is nilpotent, implies such isomorphism. Unfortunately this condition is equivalent to the condition that ω is trivial. For, either by Proposition 5 in [6] or the argument given in the proof of Proposition 1, we have that ω2nilpotent implies necessarily that ω2is trivial, since every element of Q is torsion of odd order. Using now Theorem IX.7 in [16], we conclude that the triviality of ω follows from the one of ω2.
Example 5.7 (Geometric). Finally we will show how to realize geometrically the
Example 5.6. Let us consider any action ω : Q→ Aut(Z ⊕ · · · ⊕ Z
n
). Consider the action
of Q onRn, Q× Rn→ Rngiven by the linear transformation which have matrices ω(Q). This action certainly induces an action on the torus Tn= S1× · · · × S 1
n
, which we also denote by ω. Now let M be a manifold where Q acts freely. So there is a free action
Q× Tn× M → Tn× M given by (q, x, y) → (q · x, q · y). Therefore, we have the map Tn× M → (Tn× M)/Q, where (Tn× M)/Q is the orbit space, which is a finite regular
So we get a short exact sequence 1→ π1 Tn× M→ π1 Tn× M Q → Q → 1 which is 1→ Zn⊕ π1(M)→ G → Q → 1
where G= π1((Tn× M)/Q). Now we apply the above procedure to Example 5.6. Let ω be the action given by the upper left corner 2× 2 submatrix of the 3 × 3 matrix given
in Example 5.6. Let M be the circle S1 where Z3 acts freely by rotating 120 degrees. So we get a finite cover T3→ N where the three manifold N has homology given by
Hi(N,Z2)≈ Z2for 3 i 0. Certainly Hi(T3,Z2)→ Hi(N,Z2) is not an isomorphism for all i and the result follows. Observe that Hi(N,Z2)≈ Z2 for 3 i 0 which is the same as the H∗(RP3,Z2), where RP3is the 3-projective space. It is not hard to show that the cohomology ring structures of these two spaces are different.
Acknowledgements
We would like to thank Professor D. Randall for pointing out relevant references and Professor U. Koschorke for his helpful comments. Also we are grateful to the referee for his suggestions improving substantially the presentation of this work.
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